# Number system

$\text{1.&space;Find&space;the&space;value&space;of}&space;\,&space;\,&space;98&space;\times&space;98$

A. 9604

B. 9801

C. 9980

D. 9506

Ans: A

Explanation:

$98&space;\times&space;98&space;=&space;\left&space;(&space;98&space;\right&space;)^2&space;=&space;\left&space;(&space;100&space;-&space;2&space;\right&space;)^2$

$\text{This&space;is&space;in&space;the&space;form&space;of}&space;\,\,&space;\left&space;(&space;a&space;-&space;b&space;\right&space;)^2&space;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&space;\left&space;[&space;\left&space;(&space;a&space;-&space;b&space;\right&space;)^2&space;=&space;a^2&space;+&space;b^2&space;-&space;2ab\right&space;]$

$\left&space;(&space;100&space;-&space;2&space;\right&space;)^2&space;=&space;\left&space;(&space;100&space;\right&space;)^2&space;+&space;\left&space;(&space;2&space;\right&space;)^2&space;-&space;2&space;\times&space;100&space;\times&space;2$

$=&space;10000&space;+&space;4&space;-&space;400&space;=&space;9604$

$\text{2.&space;Find&space;the&space;value&space;of}&space;\,\,&space;105&space;\times&space;105$

A. 11025

B. 12045

C. 11990

D. 12590

Ans: A

Explanation:

$105&space;\times&space;105&space;=&space;\left&space;(&space;105&space;\right&space;)^2&space;=&space;\left&space;(&space;100&space;+&space;5&space;\right&space;)^2$

$\text{This&space;is&space;in&space;the&space;form&space;of}&space;\,&space;\left&space;(&space;a&space;+&space;b&space;\right&space;)^2&space;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&space;\left&space;[&space;\left&space;(&space;a&space;+&space;b&space;\right&space;)^2&space;=&space;a^2&space;+&space;b^2&space;+&space;2ab\right&space;]$

$\left&space;(&space;100&space;+&space;5&space;\right&space;)^2&space;=&space;\left&space;(&space;100&space;\right&space;)^2&space;+&space;\left&space;(&space;5&space;\right&space;)^2&space;+&space;2&space;\times&space;100&space;\times&space;5$

$=&space;10000&space;+&space;25&space;+&space;1000&space;=&space;11025$

$\text{3.&space;Find&space;the&space;value&space;of}&space;\,\,&space;20&space;\times&space;19&space;+&space;20&space;\times&space;21$

A. 600

B. 700

C. 800

D. 900

Ans: C

Explanation:

$20&space;\times&space;19&space;+&space;20&space;\times&space;21&space;=&space;20\left&space;(&space;19&space;+&space;21&space;\right&space;)$

$=&space;20(40)$

$=&space;800$

4. Evaluate 500 – ? = 150 + 250

A. 50

B. 75

C. 90

D. 100

Ans: D

Explanation:

$\text{Let&space;500}&space;-&space;x&space;=&space;150&space;+&space;250$

$500&space;-&space;x&space;=&space;400$

$500&space;-&space;400&space;=&space;x$

$x=&space;100$

$\text&space;{&space;5.&space;Total&space;number&space;of&space;prime&space;factors&space;in&space;the&space;expression}&space;\,&space;\left&space;(&space;4&space;\right&space;)^5&space;\times&space;\left&space;(&space;9&space;\right&space;)^6&space;\times&space;\left&space;\left&space;(&space;11&space;\right&space;)^2&space;\,&space;\text{is}$

A. 20

B. 24

C. 25

D. 22

Ans: B

Explanation:

$\left&space;(&space;4&space;\right&space;)^5&space;\times&space;\left&space;(&space;9&space;\right&space;)^6&space;\times&space;\left&space;(&space;11&space;\right&space;)^2&space;=&space;\left&space;(&space;2&space;\times&space;2&space;\right&space;)^5&space;\times&space;\left&space;(&space;3&space;\times&space;3&space;\right&space;)^6&space;\times&space;\left&space;(&space;11&space;\right&space;)^2$

$=&space;2^5&space;\times&space;2^5&space;\times&space;3^6&space;\times&space;3^6&space;\times&space;11^2$

$=&space;2^{10}&space;\times&space;3^{12}&space;\times&space;11^2$

$\therefore&space;\text{Total&space;number&space;of&space;prime&space;factors&space;are&space;}&space;=&space;10&space;+&space;12&space;+&space;2&space;=&space;24$

6. 9548 + 7314 = 8362 + ?

A. 8500

B. 8000

C. 6000

D. 9000

Ans: A

Explanation:

$\text{Let}&space;\,\,&space;9548&space;+&space;7314&space;=&space;8362&space;+&space;x$

$16862&space;=&space;8362&space;+&space;x$

$16862&space;-&space;8362&space;=&space;x$

$8500&space;=&space;x$

7. The total number of even prime numbers are

A. 0

B. 1

C. 2

D. 3

Ans: B

Explanation:

$\text{One&space;and&space;only&space;even&space;prime&space;number&space;is&space;2}$

$\text{8.}&space;\,\,&space;31&space;\times&space;31&space;+&space;19&space;\times&space;19&space;+&space;2&space;\times&space;31&space;\times&space;19&space;=&space;?$

A. 1500

B. 2000

C. 1800

D. 2500

Ans: D

Explanation:

$31&space;\times&space;31&space;+&space;19&space;\times&space;19&space;+&space;2&space;\times&space;31&space;\times&space;19&space;=&space;\left&space;(&space;31&space;\right&space;)^2&space;+&space;\left&space;(&space;19&space;\right&space;)^2&space;+&space;2&space;\times&space;31&space;\times&space;19$

$=&space;\left&space;(&space;31&space;+&space;19&space;\right&space;)^2$

$=&space;\left&space;(&space;50&space;\right&space;)^2$

$=&space;2500$

$\text{9.&space;If}&space;\,\,&space;x^2&space;-&space;190&space;=&space;6&space;\,\,&space;\text{find&space;the&space;value&space;of&space;}x$

A. 10

B. 12

C. 14

D. 16

Ans: C

Explanation:

$x^2&space;-&space;190&space;=&space;6$

$x^2&space;=&space;6&space;+&space;190$

$x^2&space;=&space;196$

$x&space;=&space;\sqrt{196}&space;=&space;14$

10. Find the sum of first 10 numbers

A. 70

B. 55

C. 60

D. 95

Ans: B

Explanation:

$\text{sum&space;of&space;the&space;first&space;n&space;numbers&space;=&space;}&space;\,&space;\left&space;[&space;1&space;+&space;2&space;+&space;3&space;+&space;....&space;+&space;n&space;=&space;\frac{n\left&space;(&space;n&space;+&space;1&space;\right&space;)}{2}\right&space;]$

$\text{sum&space;of&space;the&space;first&space;10&space;numbers}&space;=&space;\frac{10\left&space;(&space;10&space;+&space;1&space;\right&space;)}{2}&space;=&space;\frac{10\left&space;(&space;11&space;\right&space;)}{2}&space;=&space;\frac{110}{2}&space;=&space;55$